To collect all variables on one side of the equation \( \frac{1}{2}(6x + 8) = x - 8 \), let's first simplify the left-hand side:
- Distribute \( \frac{1}{2} \): \[ \frac{1}{2}(6x + 8) = 3x + 4 \]
Now we have the equation: \[ 3x + 4 = x - 8 \]
Next, we will rearrange it by moving all terms involving \( x \) to one side and all constant terms to the other side. We can do this in different ways:
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Subtract \( x \) from both sides: \[ 3x - x + 4 = -8 \] This simplifies to: \[ 2x + 4 = -8 \] (Correct Answer)
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Subtract \( 4 \) from both sides: \[ 3x + 4 - 4 = x - 8 - 4 \] This simplifies to: \[ 3x = x - 12 \] (Correct Answer)
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Subtract \( 4 \) from both sides and then increase both sides by \( 8 \): \[ 3x + 4 - 4 + 8 = x - 8 + 8 \] This simplifies to: \[ 3x + 12 = x \] (Correct Answer)
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Rearranging the equation can also lead us to: \[ 3x + 4 + 8 = x \] This simplifies to: \[ 8 = -2x - 8 \] (Correct Answer, if we apply a different approach)
From your responses, the correct answers after rearranging would be:
- \( 2x + 4 = -8 \)
- \( 3x = x - 12 \)
- \( 3x + 12 = x \)
So the correct answers based on these steps are:
- \( 2x + 4 = -8 \)
- \( 3x = x - 12 \)
- \( 3x + 12 = x \)
Note: The other responses provided (like \( 4 = -2x - 8 \) and \( 8 = -2x - 8 \)) do not appear to be in a consistent arrangement based on the starting equation.